Abstract
We consider embedded compact hypersurfacesM in a halfspace of hyperbolic space with boundary∂M in the boundary geodesic hyperplaneP of the halfspace and with non-zero constant mean curvature. We prove the following. Let {M n } be a sequence of such hypersurfaces with∂M n contained in a disk of radiusr n centered at a pointσ ε P such thatr n → 0 and that eachM n is a large. H-hypersurface,H > 1. Then there exists a subsequence of {M n } converging to the sphere of mean curvatureH tangent toP atσ. In the case of smallH-hypersurfaces orH ≤ 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {M n } which are graphs. The convergence is smooth on compact subset of ℍ3 σ.
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Nelli, B., Semmler, B. Some remarks on compact constant mean curvature hypersurfaces in a halfspace of ℍn+1 . J Geom 64, 128–140 (1999). https://doi.org/10.1007/BF01229218
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DOI: https://doi.org/10.1007/BF01229218